# Is it more correct to say that the square root of 25 is equal to 5 OR -5? [duplicate]

If you were to ask me what the square root of 25 is, I would quickly say 5. But is that the correct answer?

Or is the true answer actually 5 OR -5?

In other words, does the square root of a real, positive number ALWAYS have two solutions?

There are two solutions to the equation $x^2 = 25$, namely $+5$ and $-5$.

There is only one number corresponding to $\sqrt{25}$, namely $+5$. That's because we humans, in modern mathematics, have defined $\sqrt{25}$ to mean "The positive solution to $x^2 = 25$". There is no objective, mathematical reason to pick the positive over the negative, other than convinience. But we have to make a choice, so we've gone for the positive one, because overall that choice means more readable expressions and simpler use.

You always have

$$\sqrt {25}=5$$

is is how this function is defined.

Though when you have to find a number $x$ such that:

$$x^2=25$$

there is two solutions $5$ and $-5$.

Don't forget the last one !

The square root of x is usually not defined as the solution of $y^{2}=x$ but as the positive solution for this equation. Thus it is more correct, to use the positive number, but mostly by convention.

It depends on your definition. See, in complex analysis, you have so called branch-cuts, where you define the domain of your function. You can aswell define the range of the function $\sqrt{\cdot}$ as $\mathbb{R}_{\leqslant 0}$. It does not matter. And yes, if $x$ solves $x^2 = y$ for positive $y$, so does $-x$.